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# Multiplying monomials by polynomials

Sal multiplies -4x² by (3x² + 25x - 7). Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

Multiply negative 4x squared
by the whole expression 3x squared plus 25x minus 7. So if you multiply anything
times a whole expression, you really just use the
distributive property to multiply each term
of the expression by the negative 4x squared. So we're going to
have to distribute this negative 4x squared over
every term in the expression. So first, we could
start with negative 4x squared times 3x squared. So we can write that. We're going to have negative
4x squared times 3x squared. And to that, we're going to add
negative 4x squared times 25x. And to that, we're going
to add negative 4x squared times negative 7. So let's just simplify
this a little bit. Now, we can obviously
swap the order. We're just multiplying
negative 4 times x squared times 3
times x squared. And actually, I'll
do out every step. Eventually, you can do
some of this in your head. This is the exact same thing
as negative 4 times 3 times x squared times x squared. And what is that equal to? Well, negative 4 times
3 is negative 12. And x squared times x
squared-- same base. We're taking the product. That's going to be
x to the fourth. So this right here is
negative 12x to the fourth. Now let's think about
this term over here. This is the same thing
as-- and of course, we have this plus out here. And then this part right
here is the exact same thing as 25 times negative 4
times x squared times x. So let's just multiply
the numbers out here. These were the coefficients. 25 times negative
4 is negative 100. So it'll plus negative
100, or we could just say it's minus 100. And then we have
x squared times x, or x squared times x
to the first power. Same base-- we can
add the exponents. 2 plus 1 is 3. So this is negative
100x to the third power. And then let's look at
this last term over here. We have negative 4x squared. So this is going
to be plus-- that's this plus right over here. We have negative 4. We can multiply that
times negative 7. And then multiply
that times x squared. I'm just changing the order
in which we multiply it. So negative 4 times
negative 7 is positive 28. And then I'm going to multiply
that times the x squared. There's no simplification
to do, no like terms. These are different powers of x. So we are done.